Examples of non connective C*-algebras

We present an example of two infinite families of not connective groups. Both of them are generalized of the 3-dimensional Hantzsche-Wendt group.


Introduction
For a Hilbert space H, we denote by L(H) the C ⋆ -algebra of bounded and linear operators on H. The ideal of compact operators is denoted by K ⊂ L(H). For the C ⋆ -algebra A, the cone over A is defined as CA = C 0 [0, 1)⊗A, the suspension of A as SA = C 0 (0, 1) ⊗ A. which is liftable to a completely positive and contractive map φ : A → n CL(H). * The first author is supported by the Polish National Science Center grant DEC2017/01/X/ST1/00062. For a discrete group G, we define I(G) to be the augmention ideal, i.e. the kernel of the trivial representation C ⋆ (G) → C. G is called connective if I(G) is a connective C ⋆ -algebra. From definition (see [5, p. 4921]) connectivity of G may be viewed as a stringent topological property that accounts simultaneously for the quasidiagonality of C ⋆ (G) and the verification of the Kadison-Kaplansky conjecture for certain classes of groups. Here we can referring to conjecture from 2014 [2, p. 166]. If G is a discrete, countable, torsion-free, amenable group, then the natural map 4. a torsion-free crystallographic group is connective if and only if is locally indicable if and only if is diffuse (see below) and [4].
A discrete group G is called locally indicable if every finitely generated nontrivial subgroup L of G has an infinite abelianization. The group G is called diffuse if every non-empty finite subset A of G has an element a ∈ A such that for any g ∈ G, either ga or g −1 a is not in A, [4], [8].
More examples of nonabelian connective groups were exhibited in [4], [5], [7]. The above group G 2 is a 3-dimensional, torsion-free crystallographic group, where a crystallographic group Γ, of dimension n is a cocompact and discrete subgroup of the isometry group E(n) = O(n) ⋉ R n of the Euclidean space R n . Γ is cocompact if and only if the orbit space E(n)/Γ is compact. From Bieberbach theorems (see [10,Chapter 1]) any crystallographic group Γ defines a short exact sequence where a free abelian group Z n is a maximal abelian subgroup and H is a finite group. H is sometimes called a holonomy group of Γ. The above group G 2 is isomorphic to the subgroup E(3) and is generated by A torsion-free crystallographic group is called a Bieberbach group. The orbit space R n /Γ of a Bieberbach group is a n-dimensional closed flat Riemannian manifold M with holonomy group isomorphic to H. A general characterization of connective Bieberbach groups is given in [4].
The following two theorems give us a landscape of them.
For the properties of GHW groups we refer to [10,Chapter 9]. We have G 0 = 1 and G 1 ≃ Z. Combinatorial Hantzsche-Wendt groups are torsion-free, see [1,Theorem 3.3] and for n ≥ 2 are nonunique product groups. A group G is called a unique product group if given two nonempty finite subset X, Y of G, there exists at least one element g ∈ G which has a unique representation g = xy with x ∈ X and y ∈ Y. We are ready to present our main result. Proposition 1. Generalized Hantzsche-Wendt groups with trivial center and nonabelian, combinatorial Hantzsche-Wendt groups are not connective.
Proof: From [3, Remark 2.8 (i)] the connectivity property is inherited by subgroups. Let G be any group from family of GHW groups or family of combinatorial Hanzsche-Wendt groups. In both cases a group G 2 is a subgroup of G. In the first case it follows from [10, Proposition 9.7]. In the second case it follows from definition, see [1,Prop. 3.4]. Note that in the case of GHW groups we can also use Theorem 2, since all these groups have a finite abelianizations.
Remark 1. From [11], for n ≥ 3, G n has a non-abelian free subgroup. Hence is not amenable.
Remark 2. The counterexample to the Kaplansky unit conjecture was given in 2021 by G. Gardam [9]. It was found in the group ring F 2 [G 2 ]. The Kaplansky unit conjecture states that every unit in K[G] is of the form kg for k ∈ K \ {0} and g ∈ G.